A catalytic burner is known from U.S. Pat. No. 6,358,040, which catalytic burner can generate a hydrogen-gas-containing synthesis gas from a rich fuel/air mixture during operation, and can be used as a pilot burner for a conventionally lean-operated burner of a combustion chamber of a gas turbine. By injecting a hydrogen-gas-containing synthesis gas into the burner or into a combustion space of the combustion chamber, it is possible to stabilize the homogeneous combustion reaction which takes place in the combustion space of the combustion chamber during operation. This makes it possible in particular to lower the extinguishing temperature of the combustion reaction in lean-operated burners. This makes it possible overall to reduce the combustion temperatures in the combustion space of the combustion chamber. This is particularly advantageous since the formation of nitrogen oxides increases exponentially with the reaction temperature. In order to nevertheless be able to operate the combustion chamber at higher power, the combustion chamber must be operated such that it reaches a higher outlet temperature.